Problem: I have 5 books, two of which are identical copies of the same math book (and all of the rest of the books are different).  In how many ways can I arrange them on a shelf?
Answer: There are $5!$ ways to arrange the books if they are unique, but two are identical so we must divide by $2!$ for an answer of $\dfrac{5!}{2!} = \boxed{60}$.